What is a PRP ?
A PRP is a probable prime number, a number that nobody knows how to prove
or disprove its primality.
Here are some criterions that can be used to recognize these numbers
:
- N does not have any prime factor up to 2^32.
- N cannot be trivially written into a product.
- Take 3 square-free consecutive bases such that (b1/N)=(b2/N)=(b3/N)= -1. N
should be such that b1^((N-1)/2)=b2^((N-1)/2)=b3^((N-1)/2)= -1 (mod N).
- Take a prime base such that (b/N)= -1 and b<>b1<>b2<>b3.
N should pass the strong test in base b.
- or more (if someone has got an idea) !!!
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Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
9001 | (50891^20431-1)/50890 | 96157 | Serge Batalov | 06/2022 |
9002 | (50^56597+1)/51 | 96155 | Robert Price | 03/2014 |
9003 | 22313^20374+20374^22313 | 96149 | Hans Havermann | 12/2019 |
9004 | 22279^20668+20668^22279 | 96141 | Hans Havermann | 11/2019 |
9005 | (41*10^96137-59)/9 | 96138 | Robert Price | 10/2017 |
9006 | 2^319353-110343 | 96135 | Enzo Doria | 05/2014 |
9007 | 11^92309-42 | 96130 | Lelio R Paula | 05/2016 |
9008 | (21675^22171-1)/21674 | 96129 | Serge Batalov | 06/2022 |
9009 | 14^83869-13^83869 | 96125 | Jean-Louis Charton | 01/2009 |
9010 | (7*10^96112+143)/3 | 96113 | Robert Price | 04/2016 |
9011 | 26731^3940+3940^26731 | 96112 | Hans Havermann | 07/2020 |
9012 | 22640^17583+17583^22640 | 96109 | Hans Havermann | 07/2020 |
9013 | (19178^22441-1)/19177 | 96107 | Serge Batalov | 06/2022 |
9014 | 2^319237+65835 | 96100 | Lelio R Paula | 03/2016 |
9015 | 2^319194-1841 | 96087 | Yves Minet | 03/2013 |
9016 | 23949^10280+10280^23949 | 96084 | Hans Havermann | 07/2020 |
9017 | (1927^29251-1)/1926 | 96083 | Serge Batalov | 06/2022 |
9018 | 20^73847-31 | 96078 | Lelio R Paula | 05/2016 |
9019 | (60229^20101-1)/60228 | 96075 | Serge Batalov | 06/2022 |
9020 | (28044^21601-1)/28043 | 96074 | Serge Batalov | 06/2022 |
9021 | 50^56543-49 | 96065 | Norbert Schneider | 09/2018 |
9022 | 2^319107-109555 | 96061 | Enzo Doria | 04/2014 |
9023 | (3*2^319114-1)/2627 | 96060 | Lelio R Paula | 02/2014 |
9024 | 2^319098-112157 | 96059 | Enzo Doria | 06/2014 |
9025 | 15^81673+94 | 96055 | Lelio R Paula | 07/2016 |
9026 | (85761^19471-1)/85760 | 96052 | Serge Batalov | 06/2022 |
9027 | 2^319074+1871749 | 96051 | Enzo Doria | 05/2016 |
9028 | 25670^5517+5517^25670 | 96050 | Hans Havermann | 06/2020 |
9029 | 6*10^96037+19 | 96038 | Robert Price | 11/2021 |
9030 | 2^319007-112111 | 96031 | Enzo Doria | 06/2014 |
9031 | (57987^20161-1)/57986 | 96029 | Serge Batalov | 06/2022 |
9033 | 2^318946-1421 | 96013 | Yves Minet | 01/2013 |
9032 | (68310^19861-1)/68309 | 96013 | Serge Batalov | 06/2022 |
9034 | (20294^22291-1)/20293 | 96012 | Serge Batalov | 06/2022 |
9035 | (304^38669-1)/303 | 96008 | Serge Batalov | 06/2022 |
9036 | (1840^29401-1)/1839 | 95986 | Serge Batalov | 06/2022 |
9037 | 22830^16003+16003^22830 | 95982 | Hans Havermann | 02/2020 |
9038 | 2^318834-110411 | 95979 | Enzo Doria | 05/2014 |
9039 | 2^318831-1965 | 95978 | Lelio R Paula | 03/2016 |
9040 | (5*10^95976+7)/3 | 95977 | Erik Branger | 03/2014 |
9041 | (12797^23369-1)/12796 | 95975 | Serge Batalov | 06/2022 |
9042 | (10534^23857-1)/10533 | 95963 | Serge Batalov | 06/2022 |
9043 | 22871^15696+15696^22871 | 95962 | Hans Havermann | 01/2020 |
9044 | (2^318760+858)/2 | 95957 | Stefano Morozzi | 02/2021 |
9045 | 23167^13860+13860^23167 | 95953 | Hans Havermann | 01/2020 |
9046 | (823^32911-1)/822 | 95946 | Serge Batalov | 06/2022 |
9048 | 5*3^201089-8 | 95945 | Lelio R Paula | 08/2014 |
9047 | (81783^19531-1)/81782 | 95945 | Serge Batalov | 06/2022 |
9049 | 2^318711-139 | 95942 | Lelio R Paula | 11/2012 |
9050 | 2^318635-1167 | 95919 | Lelio R Paula | 03/2016 |
9051 | 15^81555+16 | 95917 | Lelio R Paula | 07/2016 |
9052 | (37*10^95895+377)/9 | 95896 | Robert Price | 10/2017 |
9053 | (22699*2^318527+1)/21 | 95890 | Lelio R Paula | 07/2014 |
9054 | 2^318496-97305 | 95877 | Lelio R Paula | 03/2016 |
9055 | 22129^21496+21496^22129 | 95871 | Hans Havermann | 07/2020 |
9057 | 2^318466+15015 | 95868 | Lelio R Paula | 03/2016 |
9056 | 24847^7216+7216^24847 | 95868 | Hans Havermann | 07/2020 |
9058 | 23706^11065+11065^23706 | 95866 | Hans Havermann | 07/2020 |
9059 | (2863^27733-1)/2862 | 95865 | Serge Batalov | 06/2022 |
9060 | 6^123187-173129 | 95859 | Mike Oakes | 04/2011 |
9061 | 2^318423-487 | 95855 | Yves Minet | 03/2012 |
9062 | 10^95848+443709 | 95849 | Patrick De Geest | 10/2022 |
9063 | 10^95848-5000000001 | 95848 | Robert Price | 09/2022 |
9064 | 10^95848-61971 | 95848 | Patrick De Geest | 10/2022 |
9065 | (2^318359+678)/2 | 95836 | Stefano Morozzi | 09/2020 |
9066 | (10223*2^318350+1)/1563 | 95834 | Lelio R Paula | 07/2014 |
9067 | (8060^24533-1)/8059 | 95831 | Serge Batalov | 06/2022 |
9068 | (9853^23993-1)/9852 | 95814 | Serge Batalov | 06/2022 |
9070 | (2^318267+2550)/2 | 95808 | Stefano Morozzi | 09/2020 |
9069 | (5048^25873-1)/5047 | 95808 | Serge Batalov | 06/2022 |
9072 | 4*10^95799+13 | 95800 | Robert Price | 02/2021 |
9071 | (65533^19891-1)/65532 | 95800 | Serge Batalov | 06/2022 |
9073 | 45369!2+1024 | 95787 | Robert Price | 02/2015 |
9074 | 3^200758-86 | 95786 | Lelio R Paula | 05/2016 |
9075 | (2^318271+1)/(3*46031404238891*654090604244843) | 95781 | Gord Palameta | 12/2018 |
9076 | (2^318171+2030)/2 | 95779 | Stefano Morozzi | 09/2020 |
9077 | 3*10^95776-59 | 95777 | Robert Price | 10/2017 |
9078 | (82970^19471-1)/82969 | 95772 | Serge Batalov | 06/2022 |
9079 | 193^41901-2 | 95768 | Lelio R Paula | 10/2017 |
9080 | (2^318124+2502)/2 | 95765 | Stefano Morozzi | 09/2020 |
9081 | (12900^23297-1)/12899 | 95761 | Serge Batalov | 06/2022 |
9082 | 4*10^95752-71 | 95753 | Robert Price | 11/2017 |
9083 | (16955^22639-1)/16954 | 95743 | Serge Batalov | 06/2022 |
9084 | (81795^19489-1)/81794 | 95740 | Serge Batalov | 06/2022 |
9085 | (2742^27847-1)/2741 | 95737 | Serge Batalov | 06/2022 |
9086 | (6072^25303-1)/6071 | 95726 | Serge Batalov | 06/2022 |
9087 | 23640^11197+11197^23640 | 95721 | Hans Havermann | 12/2019 |
9088 | 10^95712+89329 | 95713 | Patrick De Geest | 09/2022 |
9090 | 10^95712-4000001 | 95712 | Robert Price | 07/2021 |
9089 | 10^95712-989313 | 95712 | Patrick De Geest | 09/2022 |
9091 | (60657^20011-1)/60656 | 95706 | Serge Batalov | 06/2022 |
9092 | ((10^47850-1)*10000/9+6421)*10^47850+(10^47850-1)/9 | 95704 | Hans Havermann | 05/2019 |
9093 | (509203*2^317881-1)/52805 | 95693 | Lelio R Paula | 04/2015 |
9094 | (26161^21661-1)/26160 | 95687 | Serge Batalov | 06/2022 |
9095 | (2^317853+1086)/2 | 95683 | Stefano Morozzi | 09/2020 |
9096 | 4*10^95672+91 | 95673 | Robert Price | 11/2017 |
9097 | 6*10^95665+43 | 95666 | Robert Price | 07/2020 |
9098 | (20819^22153-1)/20818 | 95663 | Serge Batalov | 06/2022 |
9099 | (64464^19891-1)/64463 | 95658 | Serge Batalov | 06/2022 |
9100 | (934^32203-1)/933 | 95652 | Serge Batalov | 06/2022 |
9101 | (11*10^95641-113)/3 | 95642 | Robert Price | 01/2017 |
9102 | (72330^19681-1)/72329 | 95632 | Serge Batalov | 06/2022 |
9103 | (13224^23203-1)/13223 | 95624 | Serge Batalov | 06/2022 |
9104 | (18211^22441-1)/18210 | 95602 | Serge Batalov | 06/2022 |
9105 | (71975^19681-1)/71974 | 95590 | Serge Batalov | 06/2022 |
9106 | (13*10^95588+11)/3 | 95589 | Robert Price | 05/2015 |
9107 | Phi(191188,10)/191189 | 95587 | Maksym Voznyy | 11/2015 |
9108 | (13368^23167-1)/13367 | 95585 | Serge Batalov | 06/2022 |
9109 | (25874^21661-1)/25873 | 95583 | Serge Batalov | 06/2022 |
9110 | 3^200325+70 | 95580 | Lelio R Paula | 05/2016 |
9111 | (25*10^95574-7)/9 | 95575 | Robert Price | 12/2014 |
9112 | (2^317485-3)/21313 | 95569 | Lelio R Paula | 06/2013 |
9113 | 2^317464+31 | 95567 | Lelio R Paula | 11/2012 |
9114 | (2^317451+2114)/2 | 95562 | Stefano Morozzi | 09/2020 |
9115 | 7^113073+82 | 95558 | Lelio R Paula | 04/2016 |
9116 | (24728^21751-1)/24727 | 95552 | Serge Batalov | 06/2022 |
9117 | (49764^20341-1)/49763 | 95536 | Serge Batalov | 06/2022 |
9118 | (36155^20959-1)/36154 | 95531 | Serge Batalov | 06/2022 |
9119 | (7725^24571-1)/7724 | 95526 | Serge Batalov | 06/2022 |
9120 | (20546^22147-1)/20545 | 95510 | Serge Batalov | 06/2022 |
9121 | 2^317251+465 | 95503 | Lelio R Paula | 03/2016 |
9122 | 19^74684+58 | 95503 | Lelio R Paula | 06/2016 |
9123 | (2^317250-7)/339 | 95500 | Lelio R Paula | 07/2013 |
9124 | 5*10^95498-21 | 95499 | Robert Price | 12/2021 |
9125 | (21615^22031-1)/21614 | 95495 | Serge Batalov | 06/2022 |
9126 | 6^122718-79 | 95494 | Lelio R Paula | 05/2016 |
9127 | 22168^20291+20291^22168 | 95485 | Hans Havermann | 10/2019 |
9128 | (77427^19531-1)/77426 | 95481 | Serge Batalov | 06/2022 |
9129 | (13026^23203-1)/13025 | 95472 | Serge Batalov | 06/2022 |
9130 | 2^317141-110491 | 95469 | Enzo Doria | 05/2014 |
9131 | (2^317097+3014)/2 | 95456 | Stefano Morozzi | 09/2020 |
9132 | 2^317094+76309 | 95455 | Enzo Doria | 05/2016 |
9133 | 2^317085+95 | 95453 | Lelio R Paula | 11/2012 |
9134 | (17*10^95451-377)/9 | 95452 | Robert Price | 11/2017 |
9135 | 2^317057-109551 | 95444 | Enzo Doria | 04/2014 |
9136 | 3^200026+28 | 95437 | Lelio R Paula | 05/2016 |
9137 | 2^317024+93 | 95434 | Donovan Johnson | 06/2008 |
9138 | (86*10^95428-437)/9 | 95429 | Robert Price | 03/2021 |
9139 | 10^95427+192591 | 95428 | Patrick De Geest | 05/2021 |
9142 | 2^316999+1605 | 95427 | Lelio R Paula | 03/2016 |
9141 | (2^317000+3210)/2 | 95427 | Stefano Morozzi | 09/2020 |
9140 | 10^95427-10001 | 95427 | Robert Price | 02/2021 |
9143 | 2^316995-110427 | 95426 | Enzo Doria | 05/2014 |
9144 | (69841^19699-1)/69840 | 95420 | Serge Batalov | 06/2022 |
9145 | (59242^19993-1)/59241 | 95415 | Serge Batalov | 06/2022 |
9147 | 22300^18987+18987^22300 | 95410 | Hans Havermann | 06/2020 |
9146 | (9478^23993-1)/9477 | 95410 | Serge Batalov | 06/2022 |
9148 | 2^316937+177 | 95408 | Lelio R Paula | 11/2012 |
9149 | 4*10^95407+91 | 95408 | Robert Price | 11/2017 |
9150 | 2^316908+1915 | 95399 | Lelio R Paula | 03/2016 |
9151 | 2^316894+99 | 95395 | Lelio R Paula | 11/2012 |
9152 | (2^316887+2546)/2 | 95393 | Stefano Morozzi | 09/2020 |
9153 | 2*10^95374-21 | 95375 | Robert Price | 09/2020 |
9154 | 6^122547+11 | 95361 | Lelio R Paula | 05/2016 |
9155 | 2^316771+82875 | 95358 | Lelio R Paula | 03/2016 |
9156 | (15609^22741-1)/15608 | 95358 | Serge Batalov | 06/2022 |
9157 | (2^316765+3102)/2 | 95356 | Stefano Morozzi | 09/2020 |
9158 | (2267^28411-1)/2266 | 95329 | Serge Batalov | 06/2022 |
9159 | (824^32693-1)/823 | 95328 | Serge Batalov | 06/2022 |
9160 | (23*10^95326+1)/3 | 95327 | Serge Batalov | 05/2009 |
9161 | 1*10^95326+57 | 95327 | Robert Price | 03/2011 |
9162 | 10^95326-393687 | 95326 | Patrick De Geest | 09/2013 |
9163 | (3^199803+4)/479539 | 95325 | Lelio R Paula | 04/2013 |
9164 | 2^316654-109481 | 95323 | Enzo Doria | 04/2014 |
9165 | 8*10^95316+51 | 95317 | Robert Price | 05/2017 |
9166 | (50^56099+43^56099)/93 | 95309 | Serge Batalov | 12/2016 |
9167 | (23*2^316559+1)/3925 | 95292 | Lelio R Paula | 09/2013 |
9169 | (71*10^95277-539)/9 | 95278 | Robert Price | 11/2017 |
9168 | 23422^11691+11691^23422 | 95278 | Hans Havermann | 10/2019 |
9170 | 19^74503-56 | 95271 | Lelio R Paula | 06/2016 |
9171 | (5038^25733-1)/5037 | 95267 | Serge Batalov | 06/2022 |
9172 | 11^91475-4380 | 95262 | Enzo Doria | 01/2013 |
9173 | (62*10^95251-611)/9 | 95252 | Robert Price | 05/2017 |
9174 | (160090^18301-1)/160089 | 95240 | Serge Batalov | 06/2022 |
9175 | (173205^18181-1)/173204 | 95238 | Serge Batalov | 06/2022 |
9176 | (158710^18313-1)/158709 | 95234 | Serge Batalov | 06/2022 |
9177 | (9979^23813-1)/9978 | 95227 | Serge Batalov | 06/2022 |
9178 | (57376^20011-1)/57375 | 95223 | Serge Batalov | 06/2022 |
9179 | (2^316321-509203)/69377 | 95218 | Lelio R Paula | 06/2013 |
9180 | 26541^3868+3868^26541 | 95216 | Hans Havermann | 07/2020 |
9181 | (79*10^95214-43)/9 | 95215 | Erik Branger | 11/2013 |
9182 | (7176^24691-1)/7175 | 95202 | Anonymous | 11/2020 |
9183 | (9*2^316233-1)/1751 | 95194 | Lelio R Paula | 02/2014 |
9184 | 5*2^316213+9 | 95191 | Lelio R Paula | 08/2014 |
9185 | 2*10^95188-91 | 95189 | Robert Price | 04/2020 |
9186 | 2^316205-1848163 | 95188 | Enzo Doria | 07/2015 |
9187 | 2^316185-2787913 | 95182 | Enzo Doria | 04/2016 |
9188 | (2^316174+3538)/2 | 95178 | Stefano Morozzi | 09/2020 |
9189 | (9787*2^316142-1)/3 | 95172 | Lelio R Paula | 04/2015 |
9190 | 2^316140-5 | 95168 | Henri Lifchitz | 02/2006 |
9191 | (15185*2^316117-1)/3 | 95165 | Lelio R Paula | 04/2015 |
9192 | (34*10^95153-7)/9/19/19157 | 95149 | Makoto Morimoto | 05/2016 |
9193 | 2^316069+76817 | 95147 | Enzo Doria | 05/2016 |
9194 | (199*10^95134-7)/3 | 95136 | Robert Price | 02/2017 |
9195 | 2^316027-97305 | 95134 | Lelio R Paula | 03/2016 |
9196 | 2^316006-2798051 | 95128 | Enzo Doria | 04/2016 |
9197 | 24994^6397+6397^24994 | 95127 | Hans Havermann | 02/2020 |
9199 | 2^315921-2797603 | 95102 | Enzo Doria | 04/2016 |
9198 | (2^315921+1314)/2 | 95102 | Stefano Morozzi | 02/2021 |
9200 | 2*10^95095+63 | 95096 | Robert Price | 10/2017 |
9201 | 2^315857+1299 | 95083 | Lelio R Paula | 03/2016 |
9202 | (2^315858+2598)/2 | 95083 | Stefano Morozzi | 09/2020 |
9204 | 14^82955-43 | 95078 | Lelio R Paula | 06/2016 |
9203 | (8*10^95078-269)/9 | 95078 | Robert Price | 09/2021 |
9205 | (23*10^95074-101)/3 | 95075 | Robert Price | 04/2021 |
9206 | 2^315827-933075 | 95074 | Paul Bourdelais | 04/2021 |
9207 | (2*10^95072-17)/3 | 95072 | Robert Price | 08/2014 |
9208 | (286*10^95059-1)/3 | 95061 | Robert Price | 11/2020 |
9209 | (2^315779+1610)/2 | 95059 | Stefano Morozzi | 09/2020 |
9210 | 2^315775-111546435 | 95058 | Paul Bourdelais | 10/2021 |
9211 | 21938^21513+21513^21938 | 95051 | Hans Havermann | 09/2020 |
9212 | 2^315727+65 | 95044 | Lelio R Paula | 11/2012 |
9213 | 2^315725+1885757 | 95043 | Enzo Doria | 03/2017 |
9214 | 23277^12058+12058^23277 | 95000 | Hans Havermann | 09/2020 |
9215 | 1*10^94983+21 | 94984 | Robert Price | 03/2011 |
9216 | 10^94983-717 | 94983 | Patrick De Geest | 08/2013 |
9217 | 17^77187-82 | 94975 | Lelio R Paula | 07/2016 |
9218 | 3*10^94964+37 | 94965 | Robert Price | 10/2017 |
9219 | (2^315481-1017)/20627285 | 94962 | Lelio R Paula | 06/2013 |
9220 | 2^315423-110329 | 94952 | Enzo Doria | 05/2014 |
9221 | 11^91169-7230 | 94943 | Enzo Doria | 02/2013 |
9222 | 7*10^94938+61 | 94939 | Robert Price | 02/2017 |
9223 | 2^315351+1025 | 94931 | Lelio R Paula | 03/2016 |
9224 | 2^315350-112253 | 94930 | Enzo Doria | 06/2014 |
9225 | (2^315343+1766)/2 | 94928 | Stefano Morozzi | 09/2020 |
9226 | (2^315343+3066)/2 | 94928 | Stefano Morozzi | 09/2020 |
9229 | ((1001327^(15817)+1)/1001328) | 94906 | Faysal YAtim | 10/2010 |
9228 | 2^315268+7 | 94906 | Lelio R Paula | 10/2012 |
9227 | 11^91133+62 | 94906 | Lelio R Paula | 05/2016 |
9230 | (2^315264+1558)/2 | 94904 | Stefano Morozzi | 09/2020 |
9231 | 30^64244+41 | 94897 | Lelio R Paula | 08/2012 |
9232 | (151*10^94885-7)/9 | 94887 | Robert Price | 07/2017 |
9233 | 4^157583+22147 | 94875 | Mike Oakes | 01/2011 |
9234 | (14*32^63026-190819)/31 | 94864 | Eric Chen | 04/2023 |
9235 | 2^315098+123 | 94854 | Lelio R Paula | 11/2012 |
9236 | 3^198791-34 | 94848 | Robert Price | 11/2013 |
9237 | 3*2^315067+7 | 94846 | Lelio R Paula | 08/2014 |
9238 | (9*2^315057+1)/73 | 94841 | Lelio R Paula | 09/2013 |
9239 | 22576^15875+15875^22576 | 94836 | Hans Havermann | 09/2020 |
9240 | (16*10^94822+53)/3 | 94823 | Robert Price | 12/2016 |
9241 | 8*6^121849-7 | 94818 | Lelio R Paula | 10/2014 |
9242 | (85*10^94802+473)/9 | 94803 | Robert Price | 03/2017 |
9243 | 21767^21852-21852^21767 | 94790 | Norbert Schneider | 06/2020 |
9244 | (2*34^61893-647)/3 | 94788 | Eric Chen | 03/2023 |
9245 | 2^314870-108761 | 94786 | Enzo Doria | 03/2014 |
9246 | 5*2^314841-3 | 94778 | Lelio R Paula | 08/2014 |
9248 | (2^314842+550)/2 | 94777 | Stefano Morozzi | 09/2020 |
9247 | 11738^23289-23289^11738 | 94777 | Norbert Schneider | 05/2022 |
9249 | (2^314787+2630)/2 | 94761 | Stefano Morozzi | 09/2020 |
9250 | 6^121772+113 | 94758 | Lelio R Paula | 05/2016 |